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Completely bounded maps between sets of Banach space operators. (English) Zbl 0747.46015
Let \(X\) and \(Y\) be Banach spaces, let \(1\leq p<\infty\), and let \(S\) be a subspace of \(B(X,Y)\), the space of all bounded operators from \(X\) into \(Y\). A norm on \(M_ n(S)\), the space of all \(n\times n\) matrices with entries in \(S\), is obtained by \(\|(a_{ij})\|_ p=\sup(\sum_ i\|\sum_ ja_{ij}x_ j\|^ p)^{1/p}\), the supremum being extended over all \(n\)-tuples \((x_ j)_{j\leq n}\) in \(X\) such that \(\sum_ j\| x_ j\|^ p\leq 1\).
If \(X_ 1\), \(Y_ 1\) are further Banach spaces, then a linear map \(u:S\to B(X_ 1,Y_ 1)\) is called \(p\)-completely bounded \((p.c.b)\) if there is a constant \(C\) such that, for any \(n\) and any \((a_{ij})\in M_ n(S)\), \(\|(u(a_{ij}))\|_ p\leq C\|(a_{ij})\|_ p\); let \(\| u\|_{pcb}\) be the least of all such \(C\)’s. In a Hilbert space situation, the case \(p=2\) corresponds to the usual definition of complete boundedness known from the theory of \(C^*\)-algebras.
Guided by a formal analogy of the definition of \(\|\cdot\|_{2cb}\) with a characterization of \(L_ 2\)-factorability of Banach space operators due to J. Lindenstrauss and A. Pełczyński [Stud. Math. 29, 275–326 (1968; Zbl 0183.40501)], the author starts by giving a rather direct proof of the following generalization of G. Wittstock’s factorization theorem [J. Funct. Anal. 40, 127–150 (1981; Zbl 0495.46005)]. Let, in the above situation, \(X=Y=H\) be a Hilbert space, and let \(u:S\to B(X_ 1,Y_ 1)\) be 2.c.b. Then there is a Hilbert space \(\hat H\), along with a representation \(\pi:B(H)\to B(H_ 1)\) and operators \(V_ 1:X_ 1\to\hat H\) and \(V_ 2:\hat H\to Y_ 1\), such that \(u(a)=V_ 2\pi(a)V_ 1\) for each \(a\in S\) and \(\| V_ 1\|\cdot\| V_ 2\|\leq\| u\|_{2cb}\), the latter inequality being then automatically an equality.
Using ultraproducts of vector valued \(L_ p\) spaces, it is possible to extend this to the general Banach space situation \(S\subset B(X,Y)\), \(1\leq p<\infty\), but the statement becomes more complicated. Nevertheless, if \(X=Y_ 1=C\), then an operator \(u\) from \(S=Y\) to \(B(X_ 1,Y_ 1)=X^*_ 1\) is p.c.b. if and only it is absolutely \(p^*\)- summing \((p^*=p/(p-1))\), and the above extension is just Pietsch’s factorization theorem for such operators. There are numerous further special results, depending on \(X,Y\) having particular properties, such as type, cotype,...

46B28 Spaces of operators; tensor products; approximation properties
46B08 Ultraproduct techniques in Banach space theory
46B07 Local theory of Banach spaces
46A32 Spaces of linear operators; topological tensor products; approximation properties
46L10 General theory of von Neumann algebras
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